Puiseux series

Definition

A Puiseux series is a type of formal power series that allows for fractional powers of the variable, enabling the study of algebraic curves and their singularities in a more flexible way. This concept is especially significant when analyzing plane curves, as it provides a means to express local behavior near points of interest, particularly at singular points where traditional power series may not suffice. By using Puiseux series, one can uncover deeper geometric properties and singularity structures that are critical in algebraic geometry.

5 Must Know Facts For Your Next Test

1. Puiseux series are often written in the form $$f(t) = ext{sum}_{n ext{ integer}} a_n t^{n/k}$$, where $$k$$ is a positive integer that indicates the fractional power.
2. They are particularly useful for studying the local properties of algebraic curves near singularities, allowing for the resolution of these points through careful analysis.
3. The coefficients of a Puiseux series can be computed through techniques like Newton's method, which helps find roots of polynomial equations.
4. Unlike traditional power series, Puiseux series can represent multi-valued functions and give rise to branches in the complex plane.
5. The convergence of a Puiseux series may not follow the usual criteria found in standard power series, making them unique in their application.

Review Questions

• How do Puiseux series differ from traditional power series, and why is this difference important in the study of plane curves?
  • Puiseux series differ from traditional power series by allowing for fractional powers of the variable, which provides a richer framework for representing functions near points of interest, especially at singularities. This flexibility is crucial when studying plane curves since traditional power series might fail to capture the intricate behaviors present at these points. By utilizing Puiseux series, mathematicians can analyze the local properties of curves more effectively and gain insights into their geometric structure.
• Discuss how Puiseux series can be used to analyze singularities of algebraic curves and what advantages they provide in this context.
  • Puiseux series are particularly valuable for analyzing singularities on algebraic curves because they can represent local behavior around these problematic points with greater precision. By allowing fractional powers, they enable the resolution of singularities, revealing the true nature of curves at these locations. This approach simplifies the understanding of intersection behavior and branching structures that are fundamental to algebraic geometry, providing a clearer picture of how curves behave near their singularities.
• Evaluate the impact of Puiseux series on the field of algebraic geometry and how they contribute to our understanding of complex curves.
  • The introduction of Puiseux series has significantly impacted algebraic geometry by providing tools for resolving singularities and understanding complex curves in a more nuanced way. Their ability to express multi-valued functions and analyze local behaviors has deepened our comprehension of curve intersections and transformations. Furthermore, Puiseux series facilitate connections between algebraic geometry and other mathematical areas such as complex analysis and topology, leading to richer theoretical developments and applications in both pure and applied mathematics.

Related terms

Algebraic Curves: Curves defined by polynomial equations in two variables, which can exhibit complex behavior such as singularities and intersections.

Power Series: An infinite series of terms that are powers of a variable, typically used to represent functions or study their properties near specific points.

Singularity: A point on a curve or surface where the mathematical object fails to be well-behaved, often leading to undefined or infinite values.